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1. Conditional Mean and Variance


  • Suppose \(Y_t\) is AR(1) series observation

  • (unconditional) Mean \[ E( Y_t ) = 0 \]

  • (unconditional) Variance \[ V( Y_t ) = \gamma(0) = (1+\phi_1^2) \sigma^2 \]



Conditional Mean

  • Conditonal Mean:
    \[ E\Big(Y_t \hspace2mm \Big| \hspace2mm \mbox{all variables realized by yesterday. }\Big) \]

  • Conditonal mean of AR(1): $Y_t = Y_{t-1} + e_t $ \[ \begin{align} E\Big(Y_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= E \Big(\phi Y_{t-1} + e_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ \\ &= \phi Y_{t-1} + E(e_t) \hspace2mm = \hspace2mm \ph Y_{t-1}. \end{align} \]

  • Conditonal variance \[ \begin{align} V\Big(Y_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= V\Big(\phi Y_{t-1} + e_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ \\ &= V(e_t) \hspace2mm = \hspace2mm \sigma^2 \end{align} \]

  • Suppose \(Y_t\) is MA(1) series observation

  • (unconditional) Mean \[ E( Y_t ) = 0 \]

  • (unconditional) Variance \[ V( Y_t ) = \gamma(0) = (1+\theta_1^2) \sigma^2 \]

  • Conditonal mean of MA(1): \(Y_t = e_t + \theta_1 e_{t-1}\) \[ \begin{align} E\Big(Y_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= E \Big( e_t + \theta_1 e_{t-1} \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) \\\\ &= E ( e_t )+ \theta_1 e_{t-1} \hspace2mm = \hspace2mm \theta_1 e_{t-1} \end{align} \] Note that \(e_{t-1}\) is not observable.

  • Conditonal variance of MA(1): \(Y_t = e_t + \theta_1 e_{t-1}\) \[ \begin{align} Var\Big(Y_t \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) \ &= Var\Big( e_t + \theta_1 e_{t-1} \hspace2mm \Big| \hspace2mm Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ &= Var( e_t ) \hspace2mm = \hspace2mm \sigma^2. \end{align} \]

  • AR(1) \[ \mbox{ Uncond'l } \hspace20mm \mbox{ cond'l } \\ E(Y_t) = 0, \hspace28mm E(Y_t|\omega_{t-1}) = \phi_1 Y_{t-1}, \\ Var(Y_t) = (1+\phi_1^2)\sigma^2 \hs10mm Var(Y_t|\omega_{t-1}) = \sigma^2 \]

  • MA(1) \[ \mbox{ Uncond'l } \hspace20mm \mbox{ cond'l } \\ E(Y_t) = 0, \hspace28mm E(Y_t|\omega_{t-1}) = \theta_1 e_{t-1}, \\ Var(Y_t) = (1+\th_1^2)\sigma^2 \hs10mm Var(Y_t|\omega_{t-1}) = \sigma^2 \]

  • For ARMA(p,q) model, conditional mean changes, but conditional variance is constant.



Financial Return

\[ \begin{align} X_t &= \mbox{ Stock Price (observation) } \\ \\ Y_t &= \ln(X_t) - \ln(X_{t-1}) \hs5mm : \mbox{ log return } \end{align} \]

## [1] "AAPL"

##   B-L test H0: the sereis is uncorrelated
##   M-L test H0: the square of the sereis is uncorrelated
##   J-B test H0: the sereis came from Normal distribution
##   SD         : Standard Deviation of the series
##      BL15  BL20  BL25 ML15 ML20 JB    SD
## [1,] 0.07 0.006 0.008    0    0  0 0.028
## [1] "GSPC"

##   B-L test H0: the sereis is uncorrelated
##   M-L test H0: the square of the sereis is uncorrelated
##   J-B test H0: the sereis came from Normal distribution
##   SD         : Standard Deviation of the series
##       BL15  BL20  BL25 ML15 ML20 JB    SD
## [1,] 0.001 0.001 0.003    0    0  0 0.007


Stylized Facts about Financial Return

  • Uncorrelated

  • Squares are correlated

  • Clustering

  • Asymmetry

  • Heavy Tailed unconditional and conditional distribution



Heteroschedasticity

Don’t confuse the conditional heteroscedasticity with (unconditonal) heteroscedasticity:




2. ARCH model

  • Engle (1985) AutoRegressive Conditionally Heteroscedastic Model \[ \begin{align} Y_t &= \sigma_t e_t \hs10mm e_t \sim_{iid} \cN(0,1) \\\\ \sigma_t^2 &= \omega+ \alpha Y_{t-1}^2 \end{align} \]

  • Mean \[ E(Y_t) = \sigma_t E(e_t) = 0 \]

  • Variance \[ V(Y_t) = V(\sigma_t)V(e_t) = \]



Conditonal Mean and Variance of ARCH

  • Conditional mean \[ E[Y_t \Big| Y_{t-1}, e_{t-1}, \ldots] \hspace3mm = \hspace3mm \sigma_t E[e_t] = 0 \]

  • Conditional variance \[ V[Y_t \Big| Y_{t-1}, e_{t-1}, \ldots] \hspace3mm = \hspace3mm \sigma_t^2 V[e_t] = \sigma_t^2 \]



ARCH is uncorrelated

  • \(Y_t\) is uncorrelated, so it will pass the Ljung-Box test.

  • But \(Y_t^2\) is correlated, so it will NOT pass the McLeod-Li test.




3. GARCH model


  • GARCH(1,1) model \[ \begin{align} Y_t &= \sigma_t e_t \hspace10mm e_t \sim_{iid} N(0,1) \\\\ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \end{align} \]



Conditional Mean and Var of GARCH

  • Conditional Mean: \(0\)

  • Conditional Variance : \(\sigma_t^2\)



Example: Daily SPY

Daily Price of SP500 ETF (SPY) from Jan 02 2000 to Dec 31 2014

##   ixDay      Date   Weekday X SPY.Open SPY.High SPY.Low SPY.Close
## 1     1  1/3/2000    Monday 1   148.25   148.25  143.88    145.44
## 2     2  1/4/2000   Tuesday 2   143.53   144.06  139.64    139.75
## 3     3  1/5/2000 Wednesday 3   139.94   141.53  137.25    140.00
## 4     4  1/6/2000  Thursday 4   139.62   141.50  137.75    137.75
## 5     5  1/7/2000    Friday 5   140.31   145.75  140.06    145.75
## 6     6 1/10/2000    Monday 6   146.25   146.91  145.03    146.25
##   SPY.Volume SPY.Adjusted
## 1    8164300       110.33
## 2    8089800       106.01
## 3   12177900       106.20
## 4    6227200       104.50
## 5    8066500       110.57
## 6    5741700       110.94

##       AIC       BIC       SIC      HQIC 
## -6.470319 -6.465360 -6.470321 -6.468556

##   B-L test H0: the sereis is uncorrelated
##   M-L test H0: the square of the sereis is uncorrelated
##   J-B test H0: the sereis came from Normal distribution
##   SD         : Standard Deviation of the series
##      BL15 BL20 BL25  ML15  ML20 JB    SD
## [1,]    0    0    0 0.719 0.855  0 1.001



Representing \(\sigma^2_t\) using \(Y_t^2\)


\[ \begin{align} \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \\\\ &= \omega + \alpha Y_{t-1}^2 + \beta \Big(\omega + \alpha Y_{t-2}^2 + \beta \sigma^2_{t-2} \Big)\\ \\ &= \omega + \beta \omega + \alpha Y_{t-1}^2 + \beta \alpha Y_{t-2}^2 + \beta^2 \sigma^2_{t-2} \\ \\ &= \omega + \beta \omega + \alpha Y_{t-1}^2 + \beta \alpha Y_{t-2}^2 + \beta^2 \Big(\omega + \alpha Y_{t-3}^2 + \beta \sigma^2_{t-3} \Big) \\ \\ &= \omega + \beta \omega + \beta^2 \omega + \alpha Y_{t-1}^2 + \beta \alpha Y_{t-2}^2 + \beta^2 \alpha Y_{t-3}^2 + \beta^3 \sigma^2_{t-3} \end{align} \]

  • Continuing, we get \[ \begin{align} &= \omega (1 + \beta + \beta^2 + \cdots ) + \alpha \sum_{i=0}^k \beta^i Y_{t-1-i}^2 + \beta^{k+1} \sigma^2_{t-1-k} \\\\ &= \frac{\omega}{1-\beta} + \alpha \sum_{i=0}^\infty \beta^i Y_{t-1-i}^2 \end{align} \]

    \[ \sigma_t^2 \hspace3mm = \hspace3mm \frac{\omega}{1-\beta} + \alpha \sum_{i=0}^\infty \beta^i Y_{t-1-i}^2 \] We will use the truncated and estimated version of this, \[ \hat \sigma_t^2 \hspace3mm = \hspace3mm \frac{\hat \omega}{1- \hat \beta} + \hat \alpha \sum_{i=0}^{t-1} \hat \beta^i Y_{t-1-i}^2 \]



Residuals of GARCH process

\[ Y_t = \sigma_t e_t \]

Using observation \(\{Y_1, \ldots, Y_n\}\), the residuals are \[ \hat e_t = Y_t/ \hat \sigma_t \]



Quasi-Maximum Likelihood Estimator


Importatnce of Conditional Distribution


\[ \begin{align} Y_t &= \sigma_t e_t \\\\ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \end{align} \]

  • [Conditonal distribution] \(=\) [Distribution of \(e_t\)]

  • Guessing correct distribution for \(e_t\) is very important in GARCH parameter estimation.



Built-in distributions



Simulation Study

True parameter (.024, .1, .8)

  • True cond’l dist: Normal Estimated using: Normal

  • True cond’l dist: std(5) Estimated using: Normal

  • True cond’l dist: sged(skew=.7, shape=1.45) Estimated using: Normal